Talk:Ring (mathematics)

From my experience, the definition of a ring that does not include associativity and the existance of a unit is the most common. Wouldn't it be advisable the encyclopedia be changed to match that definition, or are there objections to this? --- schnee

I prefer the more general definition, but I'm not sure it's the most common. Changing the definition used by Wikipedia would take quite a lot of work, as there are a great many articles which mention rings and almost all of them would need to be reworded. Even listing all articles that need to be changed (which is the necessary first step) would be a fair amount of work. In any case, there would first need to be a consensus that this is the right thing to do. --Zundark 21:08 24 May 2003 (UTC)

I agree, it would certainly be a lot of work. Who would have to be asked for a concensus on this change, though? Also, on an unrelated note, is it actually being made sure that the definitions used in the articles match the one given on this page? --- schnee

You would need to get agreement from most of those who are involved in editing the mathematics articles, particularly Axel Boldt, who has generally been the active. For myself, I feel sure that it's not usual to omit associativity, so I would be against making the change that you suggest.

As for your other question, yes we do try to make sure that Wikipedia uses consistent terminology in mathematics articles. --Zundark 10:47 25 May 2003 (UTC)

Isn't most of the interesting stuff on rings (that we'll cover) about the associative unity rings? We've mentioned the alternate usage, and any articles that wish to speak of non-associate rings should probably specify that anyway -- most of the books I've seen define rings as associate and with unit. -- Tarquin 22:40 24 May 2003 (UTC)

Most important rings are associative, but there are some exceptions (Lie algebras, Jordan algebras, the octonions). On the other hand, any book on ring theory has to cover ideals, which are non-unitary rings. --Zundark 10:47 25 May 2003 (UTC)

Personally, I'd prefer leaving the definitions as they are: for non-associative non-unitary thingies we could make algebra over a commutative ring (a module with a bilinear operation); most examples already fit under algebra over a field.

There's another issue: if we did change the definition to encompass non-unitary rings, we'd have to change the definition of "ring homomorphism" and would have to check all uses of that term to see which ones need to be changed to "homomorphism of unitary rings", since the two concepts are different. AxelBoldt 01:04 26 May 2003 (UTC)

Possible Definition Contradiction

I have an observation that may be the result of my limited knowledge of abstract algebra: the definition of rings notes that the commutative law is not an axiom of rings, but the definition states also that a ring is an abelian group. The definition of an abelian group states it is a group that is commutative. These two definitions appear to contradict each other. Can someone add some clarification? : Clif 20:05, 26 Nov 2003 (UTC)

It says that commutativity of * is not an axiom. (R,+) is abelian, so + is commutative. There is no contradiction here; these are two different operations. --Zundark 08:55, 27 Nov 2003 (UTC)

     Definition of ring :
  

When I was a graduate student in pure mathematics , the common definition of aring did not include the presense of a unity element or the requirement of the mapping of unity element onto corresponding unity element by a homomorphism, although I believe this is the accepteddefinition by the Bourbaki school.

           S. A. G.